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A136507 a(n) = Sum_{k=0..n} binomial(2^(n-k) + k, n-k).

%I A136507 #15 Sep 26 2024 03:19:34
%S A136507 1,3,10,71,1925,203904,75214965,94608676477,409763735870986,
%T A136507 6208539881584781823,334272186911271376874561,
%U A136507 64832512634295914941490910360,45811927207957062190019240099653265
%N A136507 a(n) = Sum_{k=0..n} binomial(2^(n-k) + k, n-k).
%H A136507 G. C. Greubel, <a href="/A136507/b136507.txt">Table of n, a(n) for n = 0..59</a>
%F A136507 G.f.: A(x) = Sum_{n>=0} (1 - x - 2^n*x^2)^(-1) * log(1 + 2^n*x)^n/n!.
%F A136507 a(n) ~ 2^(n^2) / n!. - _Vaclav Kotesovec_, Jul 02 2016
%F A136507 a(n) = Sum_{k=0..n} A136555(n-k+1, k). - _G. C. Greubel_, Mar 14 2021
%p A136507 A136507:= n-> add(binomial(2^k +n-k, k), k=0..n); seq(A136507(n), n=0..20); # _G. C. Greubel_, Mar 14 2021
%t A136507 Table[Sum[Binomial[2^(n-k)+k,n-k],{k,0,n}],{n,0,20}] (* _Harvey P. Dale_, Mar 08 2015 *)
%o A136507 (PARI) {a(n)=sum(k=0,n,binomial(2^(n-k)+k,n-k))}
%o A136507 for(n=0,16, print1(a(n),", "))
%o A136507 (PARI) /* a(n) = coefficient of x^n in o.g.f. series: */
%o A136507 {a(n)=polcoeff(sum(i=0,n,1/(1-x-2^i*x^2 +x*O(x^n))*log(1+2^i*x +x*O(x^n))^i/i!),n)}
%o A136507 for(n=0,16, print1(a(n),", "))
%o A136507 (Sage) [sum(binomial(2^k +n-k, k) for k in (0..n)) for n in (0..20)] # _G. C. Greubel_, Mar 14 2021
%o A136507 (Magma) [(&+[Binomial(2^k +n-k, k): k in [0..n]]): n in [0..20]]; // _G. C. Greubel_, Mar 14 2021
%Y A136507 Cf. A014070 (C(2^n, n)), A136505 (C(2^n+1, n)), A136506 (C(2^n+2, n)).
%Y A136507 Cf. A136508, A136509, A136555.
%K A136507 nonn
%O A136507 0,2
%A A136507 _Paul D. Hanna_, Jan 01 2008